## Of Gas Bills and Modulo Arithmetic

March 17, 2008

Alright, the title is meant to be catchy; there is no real connection there. Except that it is in the context of paying my apartment gas bill I found out about a simple application of modulo arithmetic in electronic transactions in the US. (I am not talking about cryptography. Nowhere near that.) While trying to pay the gas bill through an e-cheque, I found that I had to fill in something called a “routing number” before submitting the e-cheque. Having never heard of the term before, I searched a bit and found out that the number serves chiefly as an identifier of a bank (or other financial institutions) in electronic transactions. It is a 9 digit number which identifies the location of a bank and the bank. (I think a bank can have multiple routing numbers.)

The modulo arithmetic part comes in in generating and validating a routing number. To validate a routing number the following checksum is used:

[3(d1 + d4 + d7) + 7(d2 + d5 + d8) + 1(d3 + d6 + d9)] mod 10 = 0

I don’t know why this particular checksum. And I think, the the first 8 digits are known and the last digit is generated using the above formula. Not sure who does this.

[Source: wikipedia and official ABA policy (pdf link)]

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Incidentally, Gauss was responsible for modulo arithmetic (or clock arithmetic). The book The Music of the Primes has a nice description of this and many other beautiful things that Gauss developed. On this note, I also recall a quote in the brilliant book called Concrete Mathematics. In that book, the authors have included what are called “marginal notes” — they are actually the notes/comments written by the first set of students who got to read the draft of the book. The quote is – “It seems a lot of stuff is attributed to Gauss; either he was really smart or he had a great press agent.” On the same page, there is another marginal comment – “Maybe he just had a magnetic personality.”